On the maximum local mean order of sub‐k $k$‐trees of a k $k$‐tree

Abstract

AbstractFor a ‐tree , a generalization of a tree, the local mean order of sub‐‐trees of is the average order of sub‐‐trees of containing a given ‐clique. The problem whether the maximum local mean order of a tree (i.e., a 1‐tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any ‐tree , does the maximum local mean order of sub‐‐trees containing a given ‐clique occur at a ‐clique that is not a major ‐clique of ? In this paper, we give it an affirmative answer.